Exponentials and Complex Numbers

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Exponentials and Complex Numbers
Mathematics for AVT Dr Ian Drumm

• Aims To discuss exponential form of complex
  numbers and useful relations
• Learning Outcomes exponential form, Osborne's
  rule, De Moivres theorem, applications

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Review

• Cartesian Form
• Polar Form

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Consider Series
Writing x as i?
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Eulers Formula

• From previous
• Can also show

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Exponential Form

• Consider polar form
• From Eulers

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Some Identities

• Using Eulers
• But we know
• Hence

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Hyperbolic and Trigonometric Functions
Lets try substituting hyperbolic functions into
trig identities
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Osbornes Rule

• Hyperbolic function identities found by replacing

cos(?) with cosh(?) and sin(?) with
sinh(?)
Except every occurrence of sin2? is replaced by
-sinh2?
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De Moivres Theorem

• Important relation
• In exponential form

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• Directivity of speakers and microphones
• Pressures complex
• Different sources have different phases depending
  on frequency
• Resulting pressure depends on direction

Listener
rs
r
s
r-s
?
d/2
d/2
Speakers
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• Filters
• Impedances Complex
• Potential Divider
• ViR where i is current
• Vout depends on frequency of Vin